Keywords

1 Introduction and preliminaries

After the pioneering papers were written by Aumann [1] and Debreu [2], set-valued functions in Banach spaces have been developed in the last decades. We can refer to the papers by Arrow and Debreu [3], McKenzie [4], the monographs by Hindenbrand [5], Aubin and Frankowska [6], Castaing and Valadier [7], Klein and Thompson [8] and the survey by Hess [9]. The theory of set-valued functions has been much related with the control theory and the mathematical economics.

Let Y be a Banach space. We define the following:

2Y: the set of all subsets of Y;

Cb(Y): the set of all closed bounded subsets of Y;

Cc(Y): the set of all closed convex subsets of Y;

Ccb(Y): the set of all closed convex bounded subsets of Y;

Ccc(Y): the set of all closed compact subsets of Y.

We can consider the addition and the scalar multiplication on 2Y as follows:

C+C′={x+x′:x∈C,x′∈C′},λC={λx:x∈C},

where C,C′∈2Y and λ∈R. Further, if C,C′∈Cc(Y), then we denote by

C⊕C′=C+C′¯.

We can easily check that

λC+λC′=λ(C+C′),(λ+μ)C⊆λC+μC,

where C,C′∈2Y and λ,μ∈R. Furthermore, when C is convex, we obtain

(λ+μ)C=λC+μC

for all λ,μ∈R+.

For a given set C∈2Y, the distance function d(⋅,C) and the support function s(⋅,C) are, respectively, defined by

d(x,C)=inf{∥x−y∥:y∈C},x∈Y,s(x∗,C)=sup{〈x∗,x〉:x∈C},x∗∈Y∗.

For every pair C,C′∈Cb(Y), we define the Hausdorff distance between C and C′ by

h(C,C′)=inf{λ>0:C⊆C′+λBY,C′⊆C+λBY},

where BY is the closed unit ball in Y.

The following proposition is related with some properties of the Hausdorff distance.

Let (Ccb(Y),⊕,h) be endowed with the Hausdorff distance h. Since Y is a Banach space, (Ccb(Y),⊕,h) is a complete metric semigroup (see [7]). Debreu [2] proved that (Ccb(Y),⊕,h) is isometrically embedded in a Banach space as follows.

LetC(BY∗)be the Banach space of continuous real-valued functions onBY∗endowed with the uniform norm∥⋅∥u. Then the mappingj:(Ccb(Y),⊕,h)→C(BY∗), given byj(A)=s(⋅,A), satisfies the following properties:

(a)

j(A⊕B)=j(A)+j(B);

(b)

j(λA)=λj(A);

(c)

h(A,B)=∥j(A)−j(B)∥u;

(d)

j(Ccb(Y))is closed inC(BY∗)

for allA,B∈Ccb(Y)and allλ≥0.

Let f:Ω→(Ccb(Y),h) be a set-valued function from a complete finite measure space (Ω,Σ,ν) into Ccb(Y). Then f is Debreu integrable if the composition j∘f is Bochner integrable (see [10]). In this case, the Debreu integral of f in Ω is the unique element (D)∫Ωfdν∈Ccb(Y) such that j((D)∫Ωfdν) is the Bochner integral of j∘f. The set of Debreu integrable functions from Ω to Ccb(Y) will be denoted by D(Ω,Ccb(Y)). Furthermore, on D(Ω,Ccb(Y)), we define (f+g)(ω)=f(ω)⊕g(ω) for all f,g∈D(Ω,Ccb(Y)). Then we find that ((Ω,Ccb(Y)),+) is an abelian semigroup.

The stability problem of functional equations originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [13] for additive mappings and by Rassias [14] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [15] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [12, 14, 16–20]).

Let X be a set. A function d:X×X→[0,∞] is called a generalized metric on X if d satisfies

(1)

d(x,y)=0 if and only if x=y;

(2)

d(x,y)=d(y,x) for all x,y∈X;

(3)

d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.

Note that the distinction between the generalized metric and the usaul metric is that the range of the former includes the infinity.

Let (X,D) be a generalized metric space. An operator T:X→X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L≥0 such that d(Tx,Ty)≤Ld(x,y) for all x,y∈X. If the Lipschitz constant is less than 1, then the operator T is called a strictly contractive operator. We recall a fundamental result in the fixed point theory.

Let(X,d)be a complete generalized metric space and letJ:X→Xbe a strictly contractive mapping with Lipschitz constantL<1. Then for each given elementx∈X, either

d(Jnx,Jn+1x)=∞

for all nonnegative integersnor there exists a positive integern0such that

(1)

d(Jnx,Jn+1x)<∞, ∀n≥n0;

(2)

the sequence{Jnx}converges to a fixed pointy∗ofJ;

(3)

y∗is the unique fixed point ofJin the setY={y∈X∣d(Jn0x,y)<∞};

(4)

d(y,y∗)≤11−Ld(y,Jy)for ally∈Y.

In 1996, Isac and Rassias started to use the fixed point theory for the proof of stability theory of functional equations. Afterwards the stability problems of several functional equations by using the fixed point methods have been extensively investigated by a number of authors [19, 20, 23].

Set-valued functional equations have been studied by a number of authors and there are many interesting results concerning this problem (see [24–31]). In this paper, we define generalized additive set-valued functional equations and prove the Hyers-Ulam stability of generalized additive set-valued functional equations by using the fixed point method.

Throughout this paper, let X be a real vector space and Y a Banach space.

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